Physics-Informed Neural Networks with Architectural Physics Embedding for Large-Scale Wave Field Reconstruction

Large-scale wave field reconstruction requires precise solutions but faces challenges with computational efficiency and accuracy. The physics-based numerical methods like Finite Element Method (FEM) provide high accuracy but struggle with large-scale or high-frequency problems due to prohibitive computational costs. Pure data-driven approaches excel in speed but often lack sufficient labeled data for complex scenarios. Physics-informed neural networks (PINNs) integrate physical principles into machine learning models, offering a promising solution by bridging these gaps. However, standard PINNs embed physical principles only in loss functions, leading to slow convergence, optimization instability, and spectral bias, limiting their ability for large-scale wave field reconstruction. This work introduces architecture physics embedded (PE)-PINN, which integrates additional physical guidance directly into the neural network architecture beyond Helmholtz equations and boundary conditions in loss functions. Specifically, a new envelope transformation layer is designed to mitigate spectral bias with kernels parameterized by source properties, material interfaces, and wave physics. Experiments demonstrate that PE-PINN achieves more than 10 times speedup in convergence compared to standard PINNs and several orders of magnitude reduction in memory usage compared to FEM. This breakthrough enables high-fidelity modeling for large-scale 2D/3D electromagnetic wave reconstruction involving reflections, refractions, and diffractions in room-scale domains, readily applicable to wireless communications, sensing, room acoustics, and other fields requiring large-scale wave field analysis.

💡 Research Summary

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This paper addresses the long‑standing challenge of reconstructing large‑scale wave fields—particularly high‑frequency electromagnetic (EM) and acoustic fields—by proposing a novel Physics‑Embedded PINN (PE‑PINN). Traditional physics‑based solvers such as the Finite Element Method (FEM) or Finite‑Difference Time‑Domain (FDTD) deliver high accuracy but become computationally prohibitive when the domain spans many wavelengths or when the frequency is high. Pure data‑driven deep learning models can be fast but require massive labeled datasets that are difficult to obtain for large, dynamic environments. Standard Physics‑Informed Neural Networks (PINNs) improve data efficiency by embedding the governing PDE (the Helmholtz equation) and boundary conditions into the loss function, yet they suffer from three critical drawbacks: (i) spectral bias—neural networks preferentially learn low‑frequency components, making high‑frequency oscillations hard to capture; (ii) optimization instability when dealing with singular sources or sharp material discontinuities; and (iii) extremely slow convergence, especially for 3‑D Helmholtz problems.

PE‑PINN overcomes these limitations by moving physics from the loss term into the network architecture itself. The core idea is a kernel‑envelope decomposition of the complex field (E_z(\mathbf{x})): \